Most current aircraft possess a flight management system, for example of the FMS type, according to the acronym of the term “Flight Management System”. A system of FMS type is in particular described in document U.S. Pat. No. 8,498,769 (SACLE, J et al.). These systems afford an aid to navigation, through the display of information useful to the pilots, or else through the communication of flight parameters to an automatic piloting system. In particular, a system of FMS type allows a pilot or another qualified person to input, during pre-flight, a flight plan defined by a departure point, an arrival point, and a series of waypoints, usually referred to by the abbreviation WPT. All these points can be chosen from among points predefined in a navigation database, and which correspond to airports, radionavigation beacons, etc. The points can also be defined by their geographical coordinates and their altitude. The inputting of the waypoints can be done through a dedicated interface, for example a keyboard or a touchscreen, or else by transferring data from an external device. The computation of the aircraft trajectories can also be performed on ground stations and be transmitted to the aircraft through a data link.
A flight plan then consists of a succession of segments, or “legs” according to the terminology usually used in this technical field, defining the succession of waypoints, but also the type of trajectory or of maneuver to be followed to reach these points or perform the transition to the following segment. Other data can be entered into the flight management system, such as for example those which make it possible to characterize its mass and its distribution. When the aircraft is in flight, the flight management system precisely evaluates the state of the aircraft and the associated uncertainty, by centralizing the data originating from the various positioning devices, such as the satellite-based geo-positioning receiver, the radionavigation devices: for example DME, NDB and VOR, the inertial sensors, etc. A screen allows the pilots to view the current position of the aircraft, as well as the route that the aircraft follows, and the closest waypoints, all on a map background making it possible to simultaneously display other flight parameters and distinctive points. In particular, the information viewed allows the pilots to adjust flight parameters, such as heading, thrust, altitude, climb or descent rates, etc. or else simply to check the proper progress of the flight if the aircraft is piloted in an automatic manner. The computer of the flight management system makes it possible to determine an optimal flight trajectory, related to the minimization of a cost criterion. This cost criterion generally corresponds to fuel consumption, but it can also apply to travel time, to environmental considerations, or to a combination of these criteria.
The construction of a valid flight plan is subject to numerous constraints. Some of them are by nature unavoidable since they are related to physical laws (maximum speed of the aeroplane, maximum deceleration capability, etc. . . . , whereas others are related to performance criteria (for example, cruise at an altitude determined so as to have reduced fuel consumption), criteria of conformity to the procedures published by air traffic control (altitude constraint, speed constraint, time constraint, type of segment and of lateral transition), in-cabin passenger comfort criteria (for example, limit the “jerk”, that is to say an abrupt, uncomfortable acceleration experienced).
A flight plan generated by a system of FMS type is in particular constructed with the aid of a horizontal flight plan and of a vertical flight plan, and of transitions between these horizontal and vertical flight plans. The horizontal flight plan essentially contains a list of waypoints that the aeroplane will have to overfly, accompanied by the types of segments and of transitions defining the maneuvers to be followed so as to attain these points, whereas the vertical flight plan contains a list of altitudes of setpoints or constraints, as well as climb, descent and cruise segments linking together these various flight altitudes. On the basis of these flight plans, the FMS determines a lateral trajectory (also termed horizontal, corresponding to the horizontal flight plan) and a vertical trajectory (corresponding to the vertical flight plan). In the 2 axes, the trajectory is a set of geometric segments (straights, curves) joining together the elements of the flight plan. The transitions of the horizontal trajectory, for their part, make it possible to ensure that a flight plan is actually flyable, for example by defining a flyable circular arc between two successive straight segments. The transitions of the vertical trajectory make it possible to ensure that the vertical constraints and setpoints are properly complied with.
In the known systems of FMS type, the horizontal flight plans and trajectories on the one hand and vertical flight plans and trajectories on the other hand are produced separately. Initially, a horizontal trajectory is determined on the basis of the horizontal flight plan. Thereafter, a vertical trajectory is produced, on the basis of the vertical flight plan (constraints and setpoints in the vertical plane) and of the horizontal trajectory. As output of the vertical trajectory, the FMS has at its disposal the forecasts for altitude, speed, time, fuel, etc. As the turning radii of the lateral trajectory are dependent on the aeroplane altitude and speed, an iteration is performed on the flight plan and the lateral trajectory to adjust the angles of curvature (turns), thereby making it possible to obtain a flyable trajectory. This lateral trajectory having been recomputed, a new vertical trajectory must be generated. Loopbacks take place until the algorithm converges. In a general manner, the construction of the horizontal flight plan makes it possible to satisfy the constraints of trajectory segments, whereas the construction of the vertical flight plan makes it possible to satisfy the constraints pertaining to the flight domain of the aeroplane. These systems, though they make it possible to generate a flight plan which is flyable in a relatively simple manner and in a limited time, do not guarantee the optimality of the trajectory according to a criterion. Indeed, a non-optimal sequence of lateral and vertical flight phases can in particular bring a trajectory for which the optimization criterion exhibits improvement axes. A criterion for optimizing the trajectory can designate a property or a combination of properties of the trajectory to be maximized or minimized.
Document US 2010-0198433 (FORTIER, S et al.) describes a flight management system making it possible to recompute an optimal lateral trajectory in case of deviation from an initial flight plan, and to suggest a fuel-optimized lateral trajectory to an aircraft pilot.
Document U.S. Pat. No. 8,565,938 (COULMEAU, F. et al.) describes a method of vertical trajectory optimization associated with constraints and optimization parameters.
However, though the known techniques from the prior art make it possible to optimize horizontal and vertical trajectories separately, none makes it possible to apply a joint optimization. Thus, a vertical trajectory optimization can produce a change of flight phases which is unfavourable in respect of the combined horizontal-vertical trajectory. Likewise, if it is applied separately to the construction of a horizontal and vertical flight plan, a constraint in respect of the construction of the trajectory, for example of limit jerk, may produce a more unfavourable result for at least one optimization objective than if it is applied to a joint construction of horizontal and vertical flight plans.
A naive solution to this problem would be to perform several successive iterations of horizontal trajectory computation using the vertical trajectory and then of vertical trajectory computation using the horizontal trajectory so as to obtain a more and more optimized combined trajectory. However this method in no way guarantees, in the general case, convergence to the trajectory that is best optimized in a joint manner. It is moreover impossible to predict the computation time necessary to obtain an optimized trajectory, the former being related to the number of iterations necessary to satisfy a convergence-related stopping criterion. This is particularly problematic in the case of trajectory computations integrated into a piloting system, where it is desirable to compute a trajectory with a controlled duration.
In mathematical language, an optimization problem is a mathematical formalization of a search for an optimal solution to a problem, analytically or numerically. The standard mathematical formulation of an optimization problem in finite dimension comprises in particular the definition of a vector of the optimization parameters belonging to a space Rn, comprising the variables, parameters or unknowns; the definition of a function of Rn in R, the so-called cost function, cost criterion or objective function; the definition of equality and inequality constraints applying to the variables; it is also possible to define a subset X of Rn comprising the admissible values of the variables. Solving the optimization problem then consists in determining the values of the variables X which optimize (that is to say minimize or maximize) the cost function.
Optimal control problems are a subset of optimization problems, introduced by L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience 1962 ISBN 2881240771). An optimal control problem makes it possible to determine the control of a system which minimizes (or maximizes) a performance criterion, possibly under constraints.
A trajectory optimization problem can in particular be formalized as an optimal control problem, according to a specific formulation, termed a Bolza problem, in particular described by Bolza, O.: Lectures on the Calculus of Variations. Chelsea Publishing Company, 1094, available on Digital Mathematics library. 2nd edition republished in 1961, paperback in 2005, ISBN 978-1-4181-8201-4. A Bolza problem can be solved by the so-called direct schemes, described in particular by B. Dacorogna, “Direct Methods in the Calculus of Variations”, Springer-Verlag, ISBN 0-387-50491-5, or else F. Irene, G. Leoni, “Modern Methods in the Calculus of Variations: Lp Spaces”, Springer, ISN 978-0-387-35784-3.
Moreover, certain schemes make it possible to obtain information on the said constraints, in addition to an optimal trajectory according to the constraints formulated. For example, the Karush-Kuhn-Tucker parameters or conditions described in particular by H. W. Kuhn, A. Tucker, “Non linear programming”, “Proceedings of 2nd Berkeley Symposium”. Berkeley: University of California Press. pp. 481-492. MR 47303, make it possible to determine, after solving the problem, the constraints which were active or inactive, that is to say the constraints which have either made it impossible to solve the problem, or have limited the optimization of the cost function.
Although the optimization schemes, and in particular those using an optimal control problem setting, are known to make it possible to obtain the best theoretical solution to a trajectory computation problem, no practical solution based on these schemes exists today for computing an optimal trajectory in the case of a trajectory comprising several phases strung together in a way that is not predefined.
An aim of the invention is therefore to propose a method making it possible to predict a trajectory for an aircraft that jointly optimizes the horizontal and vertical flight plans, in particular by stringing together the horizontal and vertical flight phases in the most appropriate manner. Another aim of the invention is to identify, from among the various trajectory construction constraints, those which limit the joint horizontal and vertical optimization of the flight plan, so as to undertake the best balance between the various constraints and the optimization of the flight plan.
The notion of optimization designates the maximization or the minimization of an optimization criterion based on a property or a combination of properties of the trajectory. The optimization can in particular consist in predicting a trajectory which minimizes a cost criterion. A cost or optimization criterion can in particular apply to a property of the trajectory or a combination of properties of the trajectory, among which may for example be included:
Financial cost criteria, for example:                Fuel consumption;        Number of hours spent in flight (assuming that the flight personnel are paid in proportion to flight time);        
Environmental criteria, for example:                Greenhouse effect gas emissions;        Carbon dioxide emissions;        Nitrogen dioxide emissions;        Sound nuisance;        
Passenger comfort and satisfaction criteria:                Limitation of jerk;        Compliance with arrival time;        
Etc. . . .
It is also possible to optimize a cost criterion combining several elementary criteria, for example a weighted sum of the fuel consumption and of the time spent in flight, or else a cost criterion integrating financial and environmental costs.